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The Complex Propagation Constant

This document summarizes key concepts about the complex propagation constant γ for transmission lines. It defines γ as well as its real and imaginary components α and β. α represents attenuation of signals as they propagate along the transmission line, while β determines the phase shift. The wavelength λ and propagation velocity v of signals on the transmission line can be determined from α and β. Waves traveling in the positive and negative z directions along the line are represented by terms involving e^(±γz).

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Uzair Azhar
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824 views4 pages

The Complex Propagation Constant

This document summarizes key concepts about the complex propagation constant γ for transmission lines. It defines γ as well as its real and imaginary components α and β. α represents attenuation of signals as they propagate along the transmission line, while β determines the phase shift. The wavelength λ and propagation velocity v of signals on the transmission line can be determined from α and β. Waves traveling in the positive and negative z directions along the line are represented by terms involving e^(±γz).

Uploaded by

Uzair Azhar
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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1/20/2005 The Complex Propagation Constant.

doc 1/4
Jim Stiles The Univ. of Kansas Dept. of EECS
The Complex Propagation
Constant
Recall that the current and voltage along a transmission line
have the form:

0 0
0 0
0 0
z z
z z
V ( z ) V e V e
V V
I ( z ) e e
Z Z


+ +
+
+
= +
=

where Z
0
and are complex constants that describe the
properties of a transmission line. Since is complex, we can
consider both its real and imaginary components.

( R j L)( G j C )
j


+ +
+
=



where { } { } and Re Im = = . Therefore, we can write:

z j z z jBz
e e e e
+
= =
( )


Since
j z
e

=1, then
z
e

alone determines the magnitude of


z
e

.
1/20/2005 The Complex Propagation Constant.doc 2/4
Jim Stiles The Univ. of Kansas Dept. of EECS
I.E.,
z z
e e

=

.












Therefore, expresses the attenuation of the signal due to the
loss in the transmission line.

Since
z
e

is a real function, it expresses the magnitude of


z
e

only. The relative phase ( ) z of


z
e

is therefore
determined by
( ) j z j z
e e

= only (recall 1
j z
e

).

From Eulers equation:

j z j z
e e z j z

= = +
( )
cos( ) sin( )

Therefore, z represents the relative phase ( ) z of the
oscillating signal, as a function of transmission line position z.
Since phase ( ) z is expressed in radians, and z is distance (in
meters), the value must have units of :

radians

meter z
=


z
z
e


1/20/2005 The Complex Propagation Constant.doc 3/4
Jim Stiles The Univ. of Kansas Dept. of EECS
The wavelength of the signal is the distance
2
z

over which
the relative phase changes by 2 radians. So:

2 2
2 ( )- ( ) = = z z z z

= +

or, rearranging:
2
=



Since the signal is oscillating in time at rate rad sec , the
propagation velocity of the wave is:

m

2 sec sec
p
rad m
v f
rad

= = = =







where f is frequency in cycles/sec.

Recall we originally considered the transmission line current and
voltage as a function of time and position
(i.e., ( ) and ( ) v z t i z t , , ). We assumed the time function was
sinusoidal, oscillating with frequency :

{ }
{ }
j t
j t
v z t V z e
i z t I z e
=
=

( , ) Re ( )
( , ) Re ( )

1/20/2005 The Complex Propagation Constant.doc 4/4
Jim Stiles The Univ. of Kansas Dept. of EECS

Now that we know V(z) and I(z), we can write the original
functions as:

{ }
0 0
0 0
0 0
j z t j z t z z
j z t j z t z z
v z t V e e V e e
V V
i z t e e e e
Z Z


+ +
+ +
+
= +

=


( ) ( )
( ) ( )
( , ) Re
( , ) Re


The first term in each equation describes a wave propagating in
the +z direction, while the second describes a wave propagating
in the opposite (-z) direction.







Each wave has wavelength:

2

=

And velocity:

p
v

=


0
Z ,
0
( ) z j z t
V e e
+
0
j z t z
V e e
+ ( )
z

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